The elements of non-Euclidean plane geometry and trigonometry, by H. S. Carslaw.

so, 8s1 METHOD OF GEiRARD AND MANSION

1379

~ 81. I. If P, Q are any two points on OL, such that
OP<OQ<N,
and Pp, Qq are perpendicular to OA, then OPp</ OQq.
Q P q A
FIG. 96.
We know that L pPQ+ LPQq>2 right angles.
Also L OPp + L pPQ = 2 right angles.
Therefore L OPP< z PQq.
If S is the point on OL, such that OS=V and Ss is perpendicular to OA, we know that i OSs a right angle.
It follows that L OPp<L OQq<a right angle.
II. From 0 to 5, y continually increases.
Let P and Q be any two points upon OL, su'h that
OP< OQ<.
Then we know that if Pp = Qq, we must have L pPQ = L PQq,
which is impossible by (I.).
Again, if Pp > Qq, cut off pP' = qQ, and join P'Q. (Fig. 97.)
Then LpP'Q = L P'Qq.
But LPQq<a right angle.